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Metamath Proof Explorer

Theorem elmapsnd

Description: Membership in a set exponentiated to a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021)

Ref Expression
Hypotheses elmapsnd.1 
|- ( ph -> F Fn { A } )
elmapsnd.2 
|- ( ph -> B e. V )
elmapsnd.3 
|- ( ph -> ( F ` A ) e. B )
Assertion elmapsnd 
|- ( ph -> F e. ( B ^m { A } ) )

Proof

Step Hyp Ref Expression
1 elmapsnd.1 
 |-  ( ph -> F Fn { A } )
2 elmapsnd.2 
 |-  ( ph -> B e. V )
3 elmapsnd.3 
 |-  ( ph -> ( F ` A ) e. B )
4 elsni 
 |-  ( x e. { A } -> x = A )
5 4 fveq2d 
 |-  ( x e. { A } -> ( F ` x ) = ( F ` A ) )
6 5 adantl 
 |-  ( ( ph /\ x e. { A } ) -> ( F ` x ) = ( F ` A ) )
7 3 adantr 
 |-  ( ( ph /\ x e. { A } ) -> ( F ` A ) e. B )
8 6 7 eqeltrd 
 |-  ( ( ph /\ x e. { A } ) -> ( F ` x ) e. B )
9 8 ralrimiva 
 |-  ( ph -> A. x e. { A } ( F ` x ) e. B )
10 1 9 jca 
 |-  ( ph -> ( F Fn { A } /\ A. x e. { A } ( F ` x ) e. B ) )
11 ffnfv 
 |-  ( F : { A } --> B <-> ( F Fn { A } /\ A. x e. { A } ( F ` x ) e. B ) )
12 10 11 sylibr 
 |-  ( ph -> F : { A } --> B )
13 snex 
 |-  { A } e. _V
14 13 a1i 
 |-  ( ph -> { A } e. _V )
15 2 14 elmapd 
 |-  ( ph -> ( F e. ( B ^m { A } ) <-> F : { A } --> B ) )
16 12 15 mpbird 
 |-  ( ph -> F e. ( B ^m { A } ) )